In practice, the sample estimate in (REF -REF ) or (REF ) is used to test the hypothesis that two distributions are distinct, \(f\ne g\) . To set a confidence threshold for this test, one must know the statistical distribution of (REF )
under the null hypothesis that \(f=g\) . This problem has been treated in detail [1]}, [2]}. For our purposes, it is enough to know that the rms value of \(\gamma _k\)  under the hypothesis that \(f=g\)  is given by taking the expected value of (REF ), which is given by:
\(\gamma _k^{\rm noise}=\operatorname{E}[\gamma _k^2]^{1/2}=\sqrt{\frac{m+n}{mn}}\left(1-||f||_k^2\right)^{1/2}, \) 
